Reports of the Academy of Sciences of the USSR

1. Volume 164, No. 3

UDC 517.514

MATHEMATICS

Yu. S. NIKOLSKII

BOUNDARY VALUES OF FUNCTIONS FROM WEIGHTED CLASSES

(Presented by Academician I. N. Vekua on 27 V 1965)

In the present paper we consider weighted classes of functions defined in the half-space \(x_n>0\) of points \(x=(x_1,\ldots,x_n)\) of the \(n\)-dimensional space \(E_n\). The properties of the traces of these functions on the \((n-1)\)-dimensional subspace \(E_{n-1}\), defined by the equation \(x_n=0\), are studied.

Let \(r\) be a natural number; \(1<p<\infty\); \(E^+\) the upper half-space of points with \(x_n>0\), and \(\varphi=\varphi(\rho)\) \((\rho=\sqrt{x_1^2+\cdots+x_n^2})\) a positive function (weight) defined on \(E^+\); let \(f=f(x)\) be a function defined on \(E^+\) together with its generalized derivatives up to order \(r\) inclusive. By definition, the function \(f\) belongs to the class \(L_{p,\varphi}^{(r)}(E^+)\) or \(W_{p,\varphi}^{(r)}(E^+)\), if for it the finite norm

\[ \|f\|_{L_{p,\varphi}^{(r)}(E^+)} = \sum_{|k|=r} \left\| \frac{f^{(k)}(x)}{\varphi(\rho)} \right\|_{L_p(E^+)} \]

or

\[ \|f\|_{W_{p,\varphi}^{(r)}(E^+)} = \|f\|_{L_{p,\varphi}^{(r)}(E^+)} + \|f\|_{L_p(\omega)} \]

has meaning, where \(k=(k_1,\ldots,k_n)\) is an integer vector \((k_i\ge 0,\ i=1,\ldots,n)\), \(|k|=k_1+k_2+\cdots+k_n\), \(\omega\) is the intersection of \(E^+\) with the unit ball in \(E_n\) centered at the origin, and

\[ \|f\|_{L_p(\omega)} = \left\{ \int_{\omega} |f|^p\,dx \right\}^{1/p}, \qquad f^{(k)}(x) = \frac{\partial^{|k|} f(x)} {\partial x_1^{k_1}\cdots \partial x_n^{k_n}}. \]

For \(\varphi=1\), the classes \(L_{p,\varphi}^{(r)}\) and \(W_{p,\varphi}^{(r)}\) become the well-known classes \(L_p^{(r)}\) and \(W_p^{(r)}\) of S. L. Sobolev \((^1)\).

Let \(l\) be a positive non-integer number and \(l=\bar l+\alpha\), where \(\bar l\) is an integer and \(0<\alpha<1\). By definition, a function \(\psi=\psi(x_1,\ldots,x_{n-1})\) belongs to the class \(L_{p,\varphi}^{(l)}(E_{n-1})\) or \(W_{p,\varphi}^{(l)}(E_{n-1})\), if it is given on \(E_{n-1}\) together with its generalized derivatives up to order \(\bar l\) inclusive and has the finite norm

\[ \|\psi\|_{L_{p,\varphi}^{(l)}(E_{n-1})} = \sum_{|k|=\bar l} \sum_{i=1}^{n-1} \left\{ \int_{0}^{\infty} \frac{dh}{h^{1+\alpha p}} \left\| \frac{\Delta_i(\psi^{(k)},h)} {\varphi(\bar\rho+h)} \right\|_{L_p(E_{n-1})}^{p} \right\}^{1/p} \]

or

\[ \|\psi\|_{W_{p,\varphi}^{(l)}(E_{n-1})} = \|\psi\|_{L_p(\omega_*)} + \|\psi\|_{L_{p,\varphi}^{(l)}(E_{n-1})}, \]

where \(\Delta_i(\psi,h)\) \((i=1,\ldots,n-1)\) denotes the first difference of the function \(\psi=\psi(x_1,\ldots,x_{n-1})\) with step \(h\) in the variable \(x_i\); \(\bar\rho=|x_1|+\cdots+|x_{n-1}|\), and \(\omega_*=\omega E_{n-1}\). Let the weight function \(\varphi=\varphi(t)\) on \([0,\infty)\) be continuous, positive, nonincreasing, and satisfy the inequality \(\varphi(2t)\le c\varphi(t)\)

(for sufficiently large \(t\)), where \(c\) is a positive constant independent of \(t\). Then the following theorems hold.

Theorem 1. If the function \(f \in L_{p,\varphi}^{(r)}(E^+)\) or \(W_{p,\varphi}^{(r)}(E^+)\), then for it the boundary function

\[ \psi=\psi(x_1,\ldots,x_{n-1})=f\big|_{E_{n-1}} \]

has meaning, and respectively the embeddings* hold

\[ L_{p,\varphi}^{(r)}(E^+) \to L_{p,\varphi}^{(r-1/p)}(E_{n-1}),\qquad W_{p,\varphi}^{(r)}(E^+) \to W_{p,\varphi}^{(r-1/p)}(E_{n-1}). \tag{1} \]

Theorem 2. If the function \(\psi \in L_{p,\varphi}^{(r-1/p)}(E_{n-1})\), \(W_{p,\varphi}^{(r-1/p)}(E_{n-1})\), then there exists a function \(f \in L_{p,\varphi}^{(r)}(E^+)\), \(W_{p,\varphi}^{(r)}(E^+)\), defined on \(E^+\), such that

\[ f\big|_{E_{n-1}}=\psi \]

and the embeddings hold

\[ L_{p,\varphi}^{(r-1/p)}(E_{n-1}) \to L_{p,\varphi}^{(r)}(E^+),\qquad W_{p,\varphi}^{(r-1/p)}(E_{n-1}) \to W_{p,\varphi}^{(r)}(E^+). \tag{2} \]

Theorem 3. Suppose that on \(E_{n-1}\) a system of functions is given

\[ \psi_i \in L_{p,\varphi}^{(r-i-1/p)}(E_{n-1}),\; W_{p,\varphi}^{(r-i-1/p)}(E_{n-1}) \qquad (i=0,1,\ldots,r-1). \]

Then there exists a function \(f \in L_{p,\varphi}^{(r)}(E^+), W_{p,\varphi}^{(r)}(E^+)\), defined on \(E^+\), such that

\[ \partial^i f/\partial x_n^i\big|_{E_{n-1}}=\psi_i \qquad (i=0,1,\ldots,r-1) \]

and the inequality is fulfilled

\[ \|f\|_{L_{p,\varphi}^{(r)}(E^+)} \leq c\sum_{i=0}^{r-1} \|\psi_i\|_{L_{p,\varphi}^{(r-i-1/p)}(E_{n-1})} \]

and, respectively, the analogous inequality in which everywhere \(L\) is to be replaced by \(W\).

In the present paper the investigations of L. D. Kudryavtsev \((^2)\) are developed. The classes considered here are generalizations of the weighted classes \(L_{p,\alpha}^{(r)}\), \(W_{p,\alpha}^{(r)}\) introduced by him \((\varphi=(1+\rho)^\alpha)\), for which he obtained direct and inverse embedding theorems. But our embedding theorems differ from the corresponding theorems of L. D. Kudryavtsev. Along with the class \(L_{p,\alpha}^{(r)}(E^+)\) he also considers the class \(L_{p,\alpha}^{(r)}(E_1^+)\) of functions defined on the strip \(E_1^+\{x:0<x_n\leq 1\}\). Moreover, he introduces on \(E_{n-1}\) fractional classes \(\bar L_{p,\alpha}^{(r)}(E_{n-1})\) and proves that

\[ L_{p,\alpha}^{(r)}(E^+) \to \bar L_{p,\alpha}^{(r-1/p)}(E_{n-1}) \to L_{p,\alpha}^{(r)}(E_1^+). \]

In our case, however, the inverse embedding theorem completely reverses the direct theorem (see (1) and (2)). We have also succeeded in showing that the fractional classes of L. D. Kudryavtsev \(L_{p,\alpha}^{(r-1/p)}(E_{n-1})\) are not equivalent to \(L_{p,\varphi}^{(r-1/p)}(E_{n-1})\) (for \(\varphi=(1+\rho)^\alpha,\ \alpha>0\)).

Embeddings for unweighted spaces \(W_p^r\) were studied in the works \((^1,^3,^4,^{12-15})\), while embeddings of the spaces \(L_p^{(r)}\), i.e. spaces in which the norm of the elements does not include the norm of the function itself in \(L_p\), were studied in \((^{2-4},^{15})\).

* If a function \(f\) defined on \(E^+\) belongs to the normed space \(\Lambda_1\), and its trace \(\psi=f|_{E_{n-1}}\) belongs to the normed space \(\Lambda_2\) and \(\|\psi\|_{\Lambda_2}\leq c_1\|f\|_{\Lambda_1}\), then, as usual, we write \(\Lambda_1\to\Lambda_2\). Conversely, \(\Lambda_2\to\Lambda_1\) means that there exists on \(E^+\) a function \(f\in\Lambda_1\) such that \(f|_{E_{n-1}}=\psi\) and \(\|f\|_{\Lambda_1}\leq c_2\|\psi\|_{\Lambda_2}\), where \(c_1\) and \(c_2\) are constants independent of \(f,\psi\).

Theorem 4. Let the weight function be \(\varphi=\rho^{\alpha_0}\lambda(\rho)\) \((\alpha_0=(n-p)/p)\), \(1<p<\infty\), and suppose that \(\lambda(\rho)\) satisfies the condition

\[ \int_1^\infty \frac{dz}{z\chi(z)}<\infty,\qquad \chi(z)=\left\{\min_{1\le t<\infty}\left[\frac{\lambda(zt)}{\lambda(t)}\right]^p\right\}^{1/p}. \]

Then the embedding holds

\[ W_{p,\varphi}^{(r)}(E^+)\to W_{p,(1+\rho)^k\varphi}^{(r-k)}(E^+),\qquad k=0,1,\ldots,r . \]

The last theorem is analogous to the first assertion of Theorem 1 of paper \((^5)\); the classes \(W_{\rho,\varphi}^{(r)}\) considered there were defined somewhat differently than in the present paper.

Consider the differential equation

\[ L(u)=\sum_{|k|,\,|l|\le r}(-1)^{|l|}\frac{\partial^l}{\partial x^l}\bigl(a_{kl}(x)u^{(k)}(x)\bigr)=F \tag{3} \]

in the class \(\mathfrak M\) of functions \(f\) belonging to \(L_{2,\varphi}^{(r)}(E^+)\) and satisfying the boundary conditions

\[ \partial^i f/\partial x_n^i\big|_{E_{n-1}}=\psi_i\qquad (i=0,1,\ldots,r-1), \]

where \(\psi_i\in L_{2,\varphi}^{(r-i-1/2)}(E_{n-1})\) are prescribed functions. It is assumed that

\[ a_{kl}(x)=a_{lk}(x),\qquad |a_{kl}(x)|\le M^2/[(1+\rho)^{r-\min(|k|,|l|)}\varphi(\rho)]^2, \]

\[ \sum_{|k|,\,|l|\le r} a_{kl}\xi_k\xi_l\ge \frac{\lambda}{[\varphi(\rho)]^2}\sum_{|k|=r}\xi_k^2, \]

where \(\lambda>0\) does not depend on \(x\), \(\xi_k\), \(\xi_l\). The function \(F\) has the property that the norm

\[ \sigma_F=\|F\|_{L_{2}^{(r)}(E^+)} = \sup_{\substack{\|v\|_{L_{2,\varphi}^{(r)}(E^+)}\le 1\\ v\in\mathfrak M_0}} |(F,v)|, \tag{4} \]

is finite, where

\[ (F,v)=\int_{E^+}F\cdot v\,dx \]

and \(\mathfrak M_0\) is the class of functions \(f\in L_{2,\varphi}^{(r)}(E^+)\) having zero boundary functions \(\psi_i=0\) \((i=0,1,\ldots,r-1)\). In particular, if

\[ \int_{E^+}(1+\rho)^{2r}\varphi^2F^2\,dx<\infty, \]

then condition (4) is satisfied.

Relying essentially on Theorems 1, 3, and 4, we prove by the variational method the following theorems.

Theorem 5. In the class \(\mathfrak M\) there exists, and moreover is unique, a generalized solution \(u\) of equation (3).

Theorem 6. For the generalized solution \(U\) of the boundary-value problem under consideration, the estimate

\[ \|U\|_{W_{2,\varphi}^{(r)}(E^+)} \le c\left\{ \sum_{i=0}^{r-1} \|\psi_i\|_{W_{2,\varphi}^{(r-i-1/2)}(E_{n-1})} +\sigma_F \right\}, \]

holds, where \(c\) is a constant independent of the boundary functions \(\psi_i\) \((i=0,1,\ldots,r-1)\) and of the quantity \(\sigma_F\).

If one requires the functions \(a_{kl}(x)\) and \(F(x)\) to be sufficiently smooth, then from known results (see, for example, \((^{6-9})\)) it follows that the solution \(U\) has continuous partial derivatives on \(E^+\) up to order \(2r\) inclusive and becomes a classical solution of equation (3).

Theorem 7. The classical solution \(u\) of equation (3) in the class \(\mathfrak{M}\) is unique.

The proof of the last theorem is based essentially on the fact that it is shown that the class \(\mathfrak{M}_{00}\) is everywhere dense in the class \(\mathfrak{M}_0\) in the sense of the metric

\[ D(f)=\int_{E^+}\sum_{|k|=r}\left[\frac{f^{(k)}(x)}{\varphi(\rho)}\right]^2\,dx, \]

where \(\mathfrak{M}_{00}\) is the class of finite functions \(f\in\mathfrak{M}_0\).

We note that these investigations develop the works of L. D. Kudryavtsev \((^{10},\,^{11})\), in which the variational method for solving the first boundary-value problem in the case of an unbounded domain was considered for self-adjoint elliptic equations of the second order.

In conclusion I express my deep gratitude to L. D. Kudryavtsev for posing the problem and for his constant attention.

Moscow
Institute of Physics and Technology

Received
19 V 1965

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